Brownian motion with dry friction: Fokker-Planck approach

TL;DR

This paper analytically solves a Langevin equation with dry friction, deriving exact and series solutions for the propagator and velocity correlation function, advancing understanding of Brownian motion with dry friction effects.

Contribution

It provides the first exact solutions for Brownian motion with dry friction, including series representations for cases with viscous friction and external forces.

Findings

Exact propagator and correlation functions for dry friction case

Series solutions for combined dry and viscous friction

Analytical expressions with parabolic cylinder functions

Abstract

We solve a Langevin equation, first studied by de Gennes, in which there is a solid-solid or dry friction force acting on a Brownian particle in addition to the viscous friction usually considered in the study of Brownian motion. We obtain both the time-dependent propagator of this equation and the velocity correlation function by solving the associated time-dependent Fokker-Planck equation. Exact results are found for the case where only dry friction acts on the particle. For the case where both dry and viscous friction forces are present, series representations of the propagator and correlation function are obtained in terms of parabolic cylinder functions. Similar series representations are also obtained for the case where an external constant force is added to the Langevin equation.